Pencil of Straight Lines through Intersection of Two Straight Lines
Theorem
Let $u = l_1 x + m_1 y + n_1$.
Let $v = l_2 x + m_2 y + n_2$.
Let $\LL_1$ and $\LL_2$ be straight lines embedded in a cartesian plane $\CC$, expressed using the general equations:
\(\ds \LL_1: \ \ \) | \(\ds u\) | \(=\) | \(\ds 0\) | |||||||||||
\(\ds \LL_2: \ \ \) | \(\ds v\) | \(=\) | \(\ds 0\) |
The pencil of lines through the point of intersection of $\LL_1$ and $\LL_2$ is given by:
- $\set {u + k v = 0: k \in \R} \cup \set {\LL_2}$
Proof
Let $\LL$ denote an arbitrary straight line through the point of intersection of $\LL_1$ and $\LL_2$.
From Equation of Straight Line through Intersection of Two Straight Lines, $\LL$ can be given by an equation of the form:
- $u + k v = 0$
It remains to be seen that the complete pencil of lines through the point of intersection of $\LL_1$ and $\LL_2$ can be obtained by varying $k$ over the complete set of real numbers $\R$.
We have that $\LL$ can also be given by:
- $\paren {l_1 x + m_1 y + n_1} - k \paren {l_2 x + m_2 y + n_2} = 0$
That is:
- $\paren {l_1 - k l_2} x + \paren {m_1 - k m_2} y + \paren {n_1 - k n_2} = 0$
Let the slope of $\LL$ be $\tan \psi$ where $\psi$ is the angle $\LL$ makes with the $x$-axis.
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Sources
- 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {II}$. The Straight Line: $10$. Equation of a straight line through the intersection of two given lines